Naw, OP doesn't 'get it'!

First, let's get a leading example of "sad" music, the music for the death of Siegfried from Wagner's opera 'Götterdämmerung', damnation of the gods, the fourth and final opera in Wagner's Ring of the Nibelung:

     http://www.youtube.com/watch?v=L8wHteSOwW4

Yup, that sounds sad alright! Sounds like how one feels just hit with the worst loss in life.

So, why does it make people feel better?

Well, to lead up to the answer, that music is art as in the common definition the communication, interpretation of human experience, emotion. So, that music communicates the emotion of a big loss.

Now why would one want to hear that? Because the music clearly says that many other people both understand that emotion and have experienced it; so, a listener with such a loss and emotion is not alone; their loss is not nearly the first such; many others have been there before; the listener's loss is not unique and, thus, comparatively not as bad as they might have feared; so that's good news for the listener, and they feel better. That's why!